Chapter 4 Image Enhancement in the Frequency Domain
BackgroundAny function that periodically repeats itself can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series). Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and/or cosines multiplied by a weighting function (Fourier transform).
Background
The frequency domain refers to the plane of the two dimensional discrete Fourier transform of an image. The purpose of the Fourier transform is to represent a signal as a linear combination of sinusoidal signals of various frequencies.Introduction to the Fourier Transform and the Frequency Domain
The one-dimensional Fourier transform and its inverse Fourier transform (continuous case) Inverse Fourier transform: The two-dimensional Fourier transform and its inverse Fourier transform (continuous case) Inverse Fourier transform:Introduction to the Fourier Transform and the Frequency Domain
The one-dimensional Fourier transform and its inverse Fourier transform (discrete case) DTC Inverse Fourier transform:Introduction to the Fourier Transform and the Frequency Domain
Since and the fact then discrete Fourier transform can be redefinedFrequency (time) domain: the domain (values of u) over which the values of F(u) range; because u determines the frequency of the components of the transform. Frequency (time) component: each of the M terms of F(u).
Introduction to the Fourier Transform and the Frequency Domain
F(u) can be expressed in polar coordinates:R(u): the real part of F(u) I(u): the imaginary part of F(u) Power spectrum:
The One-Dimensional Fourier Transform Example
The One-Dimensional Fourier Transform Some ExamplesThe transform of a constant function is a DC value only.
The transform of a delta function is a constant.The One-Dimensional Fourier Transform Some Examples
The transform of an infinite train of delta functions spaced by T is an infinite train of delta functions spaced by 1/T.The transform of a cosine function is a positive delta at the appropriate positive and negative frequency.
The One-Dimensional Fourier Transform Some Examples
The transform of a sin function is a negative complex delta function at the appropriate positive frequency and a negative complex delta at the appropriate negative frequency.
The transform of a square pulse is a sinc function.
Introduction to the Fourier Transform and the Frequency DomainThe two-dimensional Fourier transform and its inverse Fourier transform (discrete case) DTC
Inverse Fourier transform:
u, v : the transform or frequency variables x, y : the spatial or image variablesIntroduction to the Fourier Transform and the Frequency Domain
We define the Fourier spectrum, phase anble, and power spectrum as follows:R(u,v): the real part of F(u,v) I(u,v): the imaginary part of F(u,v)
Introduction to the Fourier Transform and the Frequency Domain
Some properties of Fourier transform:The Two-Dimensional DFT and Its Inverse
The 2D DFT F(u,v) can be obtained by taking the 1D DFT of every row of image f(x,y), F(u,y), taking the 1D DFT of every column of F(u,y)(a)f(x,y) (b)F(u,y) (c)F(u,v)
The Two-Dimensional DFT and Its Inverseshift
The Two-Dimensional DFT and Its Inverse
The Property of Two-Dimensional DFT Rotation
DFTDFT
The Property of Two-Dimensional DFT Linear Combination
DFTDFT
DFT
A
B
0.25 * A + 0.75 * B
The Property of Two-Dimensional DFT Expansion
DFTDFT
A
Expanding the original image by a factor of n (n=2), filling the empty new values with zeros, results in the same DFT.
B
Two-Dimensional DFT with Different Functions
Sine waveRectangle
Its DFT
Its DFT
Two-Dimensional DFT with Different Functions
2D Gaussian functionImpulses
Its DFT
Its DFT
Filtering in the Frequency Domain
Basics of Filtering in the Frequency DomainSome Basic Filters and Their Functions
Multiply all values of F(u,v) by the filter function (notch filter): All this filter would do is set F(0,0) to zero (force the average value of an image to zero) and leave all frequency components of the Fourier transform untouched.
Some Basic Filters and Their Functions
Lowpass filterHighpass filter
Some Basic Filters and Their Functions
Correspondence between Filtering in the Spatial and Frequency DomainConvolution theorem: The discrete convolution of two functions f(x,y) and h(x,y) of size MXN is defined as Let F(u,v) and H(u,v) denote the Fourier transforms of f(x,y) and h(x,y), then Eq. (4.2-31) Eq. (4.2-32)
Correspondence between Filtering in the Spatial and Frequency Domain
:an impulse function of strength A, located at coordinates (x0,y0)
where : a unit impulse located at the origin The Fourier transform of a unit impulse at the origin (Eq. (4.2-35)) :
Correspondence between Filtering in the Spatial and Frequency Domain
Let , then the convolution (Eq. (4.2-36))Combine Eqs. (4.2-35) (4.2-36) with Eq. (4.2-31), we obtain
Correspondence between Filtering in the Spatial and Frequency DomainLet H(u) denote a frequency domain, Gaussian filter function given the equation where : the standard deviation of the Gaussian curve. The corresponding filter in the spatial domain is
Note: Both the forward and inverse Fourier transforms of a Gaussian function are real Gaussian functions.
Correspondence between Filtering in the Spatial and Frequency Domain
Correspondence between Filtering in the Spatial and Frequency Domain
One very useful property of the Gaussian function is that both it and its Fourier transform are real valued; there are no complex values associated with them. In addition, the values are always positive. So, if we convolve an image with a Gaussian function, there will never be any negative output values to deal with.There is also an important relationship between the widths of a Gaussian function and its Fourier transform. If we make the width of the function smaller, the width of the Fourier transform gets larger. This is controlled by the variance parameter 2 in the equations.These properties make the Gaussian filter very useful for lowpass filtering an image. The amount of blur is controlled by 2. It can be implemented in either the spatial or frequency domain.Other filters besides lowpass can also be implemented by using two different sized Gaussian functions.Smoothing Frequency-Domain Filters
The basic model for filtering in the frequency domain where F(u,v): the Fourier transform of the image to be smoothed H(u,v): a filter transfer function Smoothing is fundamentally a lowpass operation in the frequency domain. There are several standard forms of lowpass filters (LPF). Ideal lowpass filter Butterworth lowpass filter Gaussian lowpass filterIdeal Lowpass Filters (ILPFs)
The simplest lowpass filter is a filter that “cuts off” all high-frequency components of the Fourier transform that are at a distance greater than a specified distance D0 from the origin of the transform.The transfer function of an ideal lowpass filter where D(u,v) : the distance from point (u,v) to the center of ther frequency rectangleIdeal Lowpass Filters (ILPFs)
Ideal Lowpass Filters (ILPFs)
Ideal Lowpass FiltersIdeal Lowpass Filters Another Example
Figure 4.13 (a) A frequency-domain ILPF of radius 5. (b) Corresponding spatial filter. (c) Five impulses in the spatial domain, simulating the values of five pixels. (d) Convolution of (b) and (c) in the spatial domain.frequency
spatial
spatial
spatial
Butterworth Lowpass Filters (BLPFs) With order n
Butterworth Lowpass Filters (BLPFs) n=2 D0=5,15,30,80,and 230Butterworth Lowpass Filters (BLPFs) Spatial Representation
n=1n=2
n=5
n=20
Gaussian Lowpass Filters (FLPFs)
Gaussian Lowpass Filters (FLPFs) D0=5,15,30,80,and 230Additional Examples of Lowpass Filtering
Additional Examples of Lowpass FilteringSharpening Frequency Domain Filter
Ideal highpass filterButterworth highpass filter
Gaussian highpass filter
Highpass Filters Spatial Representations
Ideal Highpass FiltersButterworth Highpass Filters
Gaussian Highpass FiltersThe Laplacian in the Frequency Domain
Frequency domainSpatial domain
The Laplacian filter Shift the center:
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